Abstract We show that the probability of exceptional set decays exponentially for a broad class randomized algorithms approximating solutions ODEs, admitting certain error decomposition. This includes explicit and implicit Euler schemes, two-stage Runge-Kutta scheme (under inexact information). design confidence interval exact solution an IVP perform numerical experiments to illustrate theoreti...

Explicit parametric solutions are found for a nonlinear long-wave model describing steady surface waves propagating on an inviscid fluid of finite depth in the presence of a linear shear current. The exact solutions, along with an explicit parametric form of the pressure and streamfunction give a complete description of the shape of the free surface and the flow in the bulk of the fluid. The ex...

In this article, we consider the matrix equation $AXB=C$, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solutions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem $min_X |C-AXB |_F$ where $X$ is a diagonal or monomial matrix. The explici...

in this present work, the kudryashov method and the functional variable method are used to construct exact solutions of the complex kdv equation. the kudryashov method and the functional variable method are powerful methods for obtaining exact solutions of nonlinear evolution equations.

In this paper we obtain  exact solutions of the generalized Kuramoto-Sivashinsky equation, which describes manyphysical processes in motion of turbulence and other unstable process systems.    The methods used  to determine the exact solutions of the underlying equation are the Lie group analysis  and the simplest equation method. The solutions obtained are  then plotted.

New exact Grammian determinant solutions to two (3+1)-dimensional nonlinear evolution equations are derived. Extended set of sufficient conditions consisting of linear partial differential equations with variable-coefficients is presented. Moreover, a systematic analysis of linear partial differential equations is used for solving the representative linear systems. The bilinear Bäcklund transfo...

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes SDDEs are proposed, which numerical solutions bounded in q th moment ≥ 2 and converge to exact strongly any finite interval. The 1/2 order convergence rate is yielded. Furthermore, long-time asymptotic behavior...